Let $\phi$ be CNF formula with $n$ variables and $m$ clauses.
I am looking for a reduction is $\phi$ satisfiable to a problem on a planar graph $G$ with as few vertices as possible.
The majority of reductions I have seen use "crossing gadget", which replaces edge crossing by a planar graph.
So far the best reference is $|V(G)|=m^2$.
The motivation is that the treewidth of planar graphs is at most $4.9 \sqrt{|V(G)|}$ and this gives subexponential complexity for problems exponential in the treewidth like Independent Set.